Best Known (15, 15+16, s)-Nets in Base 27
(15, 15+16, 116)-Net over F27 — Constructive and digital
Digital (15, 31, 116)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- digital (4, 20, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (3, 11, 52)-net over F27, using
(15, 15+16, 172)-Net in Base 27 — Constructive
(15, 31, 172)-net in base 27, using
- 1 times m-reduction [i] based on (15, 32, 172)-net in base 27, using
- base change [i] based on digital (7, 24, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 24, 172)-net over F81, using
(15, 15+16, 334)-Net over F27 — Digital
Digital (15, 31, 334)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2731, 334, F27, 2, 16) (dual of [(334, 2), 637, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2731, 366, F27, 2, 16) (dual of [(366, 2), 701, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2731, 732, F27, 16) (dual of [732, 701, 17]-code), using
- construction XX applied to C1 = C([727,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([727,14]) [i] based on
- linear OA(2729, 728, F27, 15) (dual of [728, 699, 16]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2729, 728, F27, 15) (dual of [728, 699, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2731, 728, F27, 16) (dual of [728, 697, 17]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2727, 728, F27, 14) (dual of [728, 701, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([727,14]) [i] based on
- OOA 2-folding [i] based on linear OA(2731, 732, F27, 16) (dual of [732, 701, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(2731, 366, F27, 2, 16) (dual of [(366, 2), 701, 17]-NRT-code), using
(15, 15+16, 50958)-Net in Base 27 — Upper bound on s
There is no (15, 31, 50959)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 235 663882 226709 034904 716689 038161 135519 923249 > 2731 [i]